Joint Random Variables

Quote:

Originally Posted by RiemannManifold

Say X and Y are independent, exponentially distributed random variables - with possibly different parameters
Determine the density func. of Z = X / Y

Let $U=X$ and $V=\tfrac{X}{Y}$ .

Then we write:
$u=x=h_u(x,y)$
$v=\tfrac{x}{y}=h_v(x,y)$
$x=u=h^{-1}_x(u,v)$
$y=\tfrac{u}{v}=h^{-1}_y(u,v)$

$J=\det \begin{bmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\ & \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{bmatrix} = \det \begin{bmatrix} 1&0 \\ & \\ \dfrac{1}{v} & \dfrac{-u}{v^2} \end{bmatrix} = \dfrac{-u}{v^2} \neq 0$ WHY?

$f_{U,V}(u,v) = f_{X,Y}\large{(}h^{-1}_x(u,v), h^{-1}_y(u,v)\large{)}|J|$

Remember that independence allows us to write the jpdf as the product of marginal pdfs.

Can you take it from here?